The first main achievement of the project has been to prove a "potential" version of the Langlands correspondence in many cases. Lafforgue's construction associates to any automorphic representation (an object with many symmetries of a number theoretic nature) over an algebraic function field a Galois representation (a realization of the symmetries satisfied by numbers themselves). The dream would be to show that any Galois representation arises this way. Our "potential" theorem shows that this is indeed the case, after passage to a finite index subgroup of the Galois group.
The second main result concerns the Langlands Program over number fields, which is of more direct interest for studying diophantine equations. As part of a large (10 author) collaboration, we have proved the first potential modularity theorems for elliptic curves, and their symmetric power Galois representations, over imaginary quadratic fields. This means that many results such as the Sato--Tate conjecture, formerly known only for elliptic curves over totally real fields (such as the rational numbers), can be shown to hold in this context. In a separate work, we also prove modularity (as opposed to potential modularity) theorems for elliptic curves, showing for the first time that a positive proportion of elliptic curves over any imaginary quadratic field are modular.
Another key achievement, of a quite different nature, has been a study of the arithmetic statistics of abelian surfaces. These are a 2-dimensional analogue of the 1-dimensional elliptic curves, which are among some of the most famous objects in diophantine (or arithmetic) geometry. We prove a result about the average size of the 3-Selmer group of an abelian surface. The definition of the Selmer group is technical, but our results have the concrete consequence that a positive density of monic degree 5 polynomials with rational coefficients take no rational square values. The technical heart of our work uses ideas from algebraic geometry that have recently played an important role in the Langlands program (in particular, in the work of Ngo on the fundamental lemma and in Lafforgue's work).
The final and most significant result we have obtained is the existence of all symmetric power liftings for holomorphic modular forms. This can be seen as the most important special case of Langlands' functoriality conjectures, an important part of the Langlands Program. We achieve this by combining results in the theory of p-adic automorphic forms, in particular the theory of eigenvarieties, with results on Selmer groups that rely in an essential way on Lafforgue's notion of pseudocharacter. This breakthrough work has been recognised by a number of major prizes.
Aside from the usual lectures in at academic seminar and conferences, the results of this project have been the subject of lectures at major conferences (including the ICM and the CDM series at Harvard), a video interview, a podcast, and articles in popular scientific magazines.
